| L(s) = 1 | − 2.76·2-s + 3-s + 5.64·4-s + 4.23·5-s − 2.76·6-s − 7-s − 10.0·8-s + 9-s − 11.7·10-s − 2.70·11-s + 5.64·12-s − 6.67·13-s + 2.76·14-s + 4.23·15-s + 16.5·16-s − 0.826·17-s − 2.76·18-s + 3.05·19-s + 23.9·20-s − 21-s + 7.46·22-s + 0.807·23-s − 10.0·24-s + 12.9·25-s + 18.4·26-s + 27-s − 5.64·28-s + ⋯ |
| L(s) = 1 | − 1.95·2-s + 0.577·3-s + 2.82·4-s + 1.89·5-s − 1.12·6-s − 0.377·7-s − 3.56·8-s + 0.333·9-s − 3.70·10-s − 0.814·11-s + 1.62·12-s − 1.85·13-s + 0.738·14-s + 1.09·15-s + 4.13·16-s − 0.200·17-s − 0.651·18-s + 0.701·19-s + 5.34·20-s − 0.218·21-s + 1.59·22-s + 0.168·23-s − 2.05·24-s + 2.59·25-s + 3.61·26-s + 0.192·27-s − 1.06·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.198366140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.198366140\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 + T \) |
| good | 2 | \( 1 + 2.76T + 2T^{2} \) |
| 5 | \( 1 - 4.23T + 5T^{2} \) |
| 11 | \( 1 + 2.70T + 11T^{2} \) |
| 13 | \( 1 + 6.67T + 13T^{2} \) |
| 17 | \( 1 + 0.826T + 17T^{2} \) |
| 19 | \( 1 - 3.05T + 19T^{2} \) |
| 23 | \( 1 - 0.807T + 23T^{2} \) |
| 29 | \( 1 - 3.57T + 29T^{2} \) |
| 31 | \( 1 + 0.0466T + 31T^{2} \) |
| 37 | \( 1 + 11.6T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 - 3.44T + 43T^{2} \) |
| 47 | \( 1 - 6.58T + 47T^{2} \) |
| 53 | \( 1 + 6.86T + 53T^{2} \) |
| 59 | \( 1 - 7.92T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 + 5.61T + 67T^{2} \) |
| 71 | \( 1 - 16.2T + 71T^{2} \) |
| 73 | \( 1 - 3.18T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 + 1.35T + 83T^{2} \) |
| 89 | \( 1 - 6.57T + 89T^{2} \) |
| 97 | \( 1 - 7.76T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.760711527769057116950842781628, −7.78755194812050359356169691794, −7.17986565899459464418882766228, −6.63618830810783519065504066081, −5.72262404277380516331990752310, −5.07147716710026926498258995721, −3.07043855544790414636902826018, −2.41576243781859408303231882185, −2.03705876252700918462196167351, −0.792226183484678420963000131039,
0.792226183484678420963000131039, 2.03705876252700918462196167351, 2.41576243781859408303231882185, 3.07043855544790414636902826018, 5.07147716710026926498258995721, 5.72262404277380516331990752310, 6.63618830810783519065504066081, 7.17986565899459464418882766228, 7.78755194812050359356169691794, 8.760711527769057116950842781628
